In Chekhov's settings there is little to no difference in the first approximation: both в кухне and на кухне mean location in a room or set of rooms used as a kitchen to cook food and maybe do the related stuff.
However, в кухне refers to the kitchen room itself, while на кухне can point to the kitchen environment in a more general sense, not necessarily a room. It would be wrong to say в кухне when you talk about a dedicated kitchen area as a part of a living room, but you can still say я был на кухне talking about that kitchen area. However this kind of floor plan may be a common thing today, but hard to imagine in the time and settings of Checkhov's novels.
Because of this room/environment difference, there is a (possibly arguable) thing that на кухне is more likely to indicate the process of making food too, so на кухне could be used to indicate that the setting was related to the food-making activities, not just kitchen premises.
There are other subtle things that may help you with the nuances, though they aren't directly related to your excerpts.
When talking about kitchen stuff of 5 people you can say на кухне работает 5 человек, but not в кухне работает 5 человек - the last one would mean that 5 people are working there right now and it's also possible that there is something different they are doing, like painting the walls.
And talking about cuisines you use в кухне, but not на кухне: чеснок популярен в кухне многих стран.